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On Coxeter algebraic varieties

The geometry of \(CB_{n}\) quartics

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Abstract

We give a new method relying on Coxeter chambers for the geometrical description of real algebraic varieties invariant under the \(CB_{n}\)-Coxeter group. It turns out that the maximal number of connected components that a \(CB_{n}\)-quartic algebraic variety can achieve is \(2^{n}+1\) for specific coefficients. Our approach establishes a deep connection between the construction of \(CB_{n}\)-polynomials using partitions of integers and the geometrical aspect of the corresponding algebraic varieties.

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  1. Sorbonne Université, 4, Place Jussieu, 75005, Paris, France

    N. C. Combe

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  1. N. C. Combe
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Correspondence to N. C. Combe.

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The study was supported by a grant from the Labex Archimede and of the A*MIDEX project (ANR-11-IDEX-0001-02), funded by the “Investissements d’Avenir” French Government programme managed by the French National Research Agency (ANR). First of all I wish to thank the referee for many interesting remarks and suggestions. I would like to thank Professor Norbert A’Campo for interesting discussions on this subject. I would like to express all my gratitude to Professor Hanna Nencka for many valuable discussions and comments. I also wish to thank Marco Robalo for advice and valuable suggestions.

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Combe, N.C. On Coxeter algebraic varieties. Math Semesterber 66, 73–87 (2019). https://doi.org/10.1007/s00591-018-0221-z

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